1 / 12

Announcements: Pass in Homework 5 now. Term project groups and topics due by Friday

DTTF/NB479: Dszquphsbqiz Day 22. Announcements: Pass in Homework 5 now. Term project groups and topics due by Friday Can use discussion forum to find teammates HW6 posted, due date next week Questions? This week: Primality testing, factoring Discrete Logs. 1.

wadkins
Download Presentation

Announcements: Pass in Homework 5 now. Term project groups and topics due by Friday

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. DTTF/NB479: Dszquphsbqiz Day 22 • Announcements: • Pass in Homework 5 now. • Term project groups and topics due by Friday • Can use discussion forum to find teammates • HW6 posted, due date next week • Questions? • This week: • Primality testing, factoring • Discrete Logs

  2. 1 The Square Root Compositeness Theorem gives a way to factor certain composite numbers Given integers n, x, and y: Then n is composite, and gcd(x-y, n) is a non-trivial factor Proof: on board Toy example showing 21 is composite using x=2 and y=16.

  3. The Miller-Rabin Compositeness Test just reorders the Fermat test’s powermod to catch pseudoprimes ? Observe: n is odd and n>1 Trick: write n-1=2km, where k >=1 ? b0 We’ll compute powers from inside out, checking if the result is +1 or -1 at each step

  4. 2 It uses the Square Root Compositeness Theorem to catch most pseudoprimes k Given odd n>1, write n-1=2km, where k >=1. Choose a base a randomly (or just pick a=2) Let b0=am(mod n) If b0=+/-1, stop. n is probably prime by Fermat For i = 1..k-1 Compute bi=bi-12. If bi=1(mod n), stop. n is composite by SRCT, and gcd(bi-1-1,n) is a factor. If bi=-1(mod n), stop. n is probably prime by Fermat. If bk=1 (mod n), stop. n is composite by SRCT Else n is composite by Fermat. b0 b0 b1 bk

  5. 3-4 Examples of Miller-Rabin • n=189 • n=561 (recall Fermat says prob prime) • Complete the table on your quiz Given odd n>1, write n-1=2km, where k >=1. Choose a base a randomly (or just pick a=2) Let b0=am(mod n) If b0=+/-1, stop. n is probably prime by Fermat For i = 1..k-1 Compute bi=bi-12. If bi=1(mod n), stop. n is composite by SRCT, and gcd(bi-1-1,n) is a factor. If bi=-1(mod n), stop. n is probably prime by Fermat. If bk=1 (mod n), stop. n is composite by SRCT Else n is composite by Fermat.

  6. Fermat’s contrapositive is OK, but Miller-Rabin is better! n Even? no div by other small primes? no yes Prime by Factoring/advanced techn.? yes prime

  7. Fermat’s contrapositive is OK, but Miller-Rabin is better! n • Finding large probable primes • #primes < x = Density of primes: ~1/ln(x) For 100-digit numbers, ~1/230. So ~1/115 of odd 100-digit numbers are prime Can start with a random large odd number and iterate, applying M-R to remove composites. We’ll soon find one that is a likely prime. Can repeat with different bases to improve probability that it’s prime. Maple’s nextprime() appears to do this, but also runs the Lucas test: http://www.mathpages.com/home/kmath473.htm Even? no div by other small primes? no Pass M-R? yes Prime by Factoring/advanced techn.? yes prime

  8. Using within a primality testing scheme n • Finding large probable primes • #primes < x = Density of primes: ~1/ln(x) For 100-digit numbers, ~1/230. So ~1/115 of odd 100-digit numbers are prime Can start with a random large odd number and iterate, applying M-R to remove composites. We’ll soon find one that is a likely prime. Can repeat with different bases to improve probability that it’s prime. Maple’s nextprime() appears to do this, but also runs the Lucas test: http://www.mathpages.com/home/kmath473.htm Odd? no div by other small primes? no Pass M-R? yes Prime by Factoring/advanced techn.? yes prime

  9. Factoring • If you are trying to factor n=pq and know that p~q, use Fermat factoring: • Compute n + 12, n + 22, n + 32, until you reach a perfect square, say r2 = n + k2 • Then n = r2 - k2 = (r+k)(r-k) • Example: factor 2405597 • The moral of the story? • Choose p and q such that _____

  10. (p-1) Algorithm • Useful if p|n and (p-1) has only small factors • Choose any a>1 (like a=2) and bound B • Compute b=aB!(mod n) (How?) • Then compute d=gcd(b-1, n) • If 1<d<n, then d is a non-trivial factor • Matlab example: n=5183. We’ll use a=2, B=6. • Why does it work?

  11. Moral of this story? • To get a 100-digit number n=pq resistant to this attack: • Make sure (p-1) has at least 1 large prime factor: • Pick p0 = nextprime(1040) • Choose k~1060 such that p=(kp0+1)is prime • How to test? • Repeat for q.

  12. Example Factor n = 3837523 • Concepts we will learn also apply to factoring really big numbers. They are the basis of the best current methods • All you have to do to win $30,000 is factor a 212 digit number. • This is the RSA Challenge: http://www.rsa.com/rsalabs/node.asp?id=2093#RSA704

More Related